Кіріспе
Нормалды үлестірілімге қатысты ықтималдықтар кестесі
Статистикада стандартты нормалды кесте, сондай-ақ бірлік нормалды кесте немесе Z кестесі деп аталатын бұл кесте, нормалды үлестірілімнің жиынтық үлестірілім функциясы Φ-ның мәндері үшін жасалған математикалық кесте. Ол стандартты нормалды үлестірілімде немесе одан әрі, кез келген нормалды үлестірілімде статистикалық көрсеткіштің белгілі бір мәннен төмен, жоғары немесе аралықта кездесу ықтималдығын анықтау үшін қолданылады. Ықтималдық кестелерін әр нормалды үлестірілім үшін басып шығару мүмкін емес, себебі нормалды үлестірілімнің сансыз көп түрлері бар. Сондықтан, нормалды үлестірілімді стандартты нормалды үлестірілімге (z-мәні деп аталады) түрлендіріп, содан кейін стандартты нормалды кесте арқылы ықтималдықтарды табу әдеттегі практика болып табылады.
Қалыпты және стандартты қалыпты үлестірім
Нормалды үлестірулер – симметриялық, қоңырау пішіндес үлестірулер, олар нақты өмірдегі деректерді сипаттауға пайдалы. Орташа мәні 0 және стандартты қатесі 1 болатын, Z әрпімен белгіленетін стандартты нормалды үлестірім – бұл нормалды үлестірудің бір түрі.
Жиынтық (Z-ден төмен)
Бұл кесте статистиканың Z-ден төмен болу ықтималдығын көрсетеді (яғни, минус шексіздіктен Z-ға дейін). z −0.00 −0.01 −0.02 −0.03 −0.04 −0.05 −0.06 −0.07 −0.08 −0.09 4.00.000030.000030.000030.000030.000030.000030.000020.000020.000020.00002 3.90.000050.000050.000040.000040.000040.000040.000040.000040.000030.00003 3.80.000070.000070.000070.000060.000060.000060.000060.000050.000050.00005 3.70.000110.000100.000100.000100.000090.000090.000080.000080.000080.00008 3.60.000160.000150.000150.000140.000140.000130.000130.000120.000120.00011 3.50.000230.000220.000220.000210.000200.000190.000190.000180.000170.00017 −3.40.000340.000320.000310.000300.000290.000280.000270.000260.000250.00024 −3.30.000480.000470.000450.000430.000420.000400.000390.000380.000360.00035 −3.20.000690.000660.000640.000620.000600.000580.000560.000540.000520.00050 −3.10.000970.000940.000900.000870.000840.000820.000790.000760.000740.00071 −3.00.001350.001310.001260.001220.001180.001140.001110.001070.001040.00100 −2.90.001870.001810.001750.001690.001640.001590.001540.001490.001440.00139 −2.80.002560.002480.002400.002330.002260.002190.002120.002050.001990.00193 −2.70.003470.003360.003260.003170.003070.002980.002890.002800.002720.00264 −2.60.004660.004530.004400.004270.004150.004020.003910.003790.003680.00357 −2.50.006210.006040.005870.005700.005540.005390.005230.005080.004940.00480 −2.40.008200.007980.007760.007550.007340.007140.006950.006760.006570.00639 −2.30.010720.010440.010170.009900.009640.009390.009140.008890.008660.00842 −2.20.013900.013550.013210.012870.012550.012220.011910.011600.011300.01101 −2.10.017860.017430.017000.016590.016180.015780.015390.015000.014630.01426 −2.00.022750.022220.021690.021180.020680.020180.019700.019230.018760.01831 −1.90.028720.028070.027430.026800.026190.025590.025000.024420.023850.02330 −1.80.035930.035150.034380.033620.032880.032160.031440.030740.030050.02938 −1.70.044570.043630.042720.041820.040930.040060.039200.038360.037540.03673 −1.60.054800.053700.052620.051550.050500.049470.048460.047460.046480.04551 −1.50.066810.065520.064260.063010.061780.060570.059380.058210.057050.05592 −1.40.080760.079270.077800.076360.074930.073530.072150.070780.069440.06811 −1.30.096800.095100.093420.091760.090120.088510.086920.085340.083790.08226 −1.20.115070.113140.111230.109350.107490.105650.103830.102040.100270.09853 −1.10.135670.133500.131360.129240.127140.125070.123020.121000.119000.11702 −1.00.158660.156250.153860.151510.149170.146860.144570.142310.140070.13786 −0.90.184060.181410.178790.176190.173610.171060.168530.166020.163540.16109 −0.80.211860.208970.206110.203270.200450.197660.194890.192150.189430.18673 −0.70.241960.238850.235760.232700.229650.226630.223630.220650.217700.21476 −0.60.274250.270930.267630.264350.261090.257850.254630.251430.248250.24510 −0.50.308540.305030.301530.298060.294600.291160.287740.284340.280960.27760 −0.40.344580.340900.337240.333600.329970.326360.322760.319180.315610.31207 −0.30.382090.378280.374480.370700.366930.363170.359420.355690.351970.34827 −0.20.420740.416830.412940.409050.405170.401290.397430.393580.389740.38591 −0.10.460170.456200.452240.448280.444330.440380.436440.432510.428580.42465 −0.00.500000.496010.492020.488030.484050.480060.476080.472100.468120.46414 z + 0.00 + 0.01 + 0.02 + 0.03 + 0.04 + 0.05 + 0.06 + 0.07 + 0.08 + 0.09
z + 0.00 + 0.01 + 0.02 + 0.03 + 0.04 + 0.05 + 0.06 + 0.07 + 0.08 + 0.09 0.00.500000.503990.507980.511970.515950.519940.523920.527900.531880.53586 0.10.539830.543800.547760.551720.555670.559620.563600.567490.571420.57535 0.20.579260.583170.587060.590950.594830.598710.602570.606420.610260.61409 0.30.617910.621720.625520.629300.633070.636830.640580.644310.648030.65173 0.40.655420.659100.662760.666400.670030.673640.677240.680820.684390.68793 0.50.691460.694970.698470.701940.705400.708840.712260.715660.719040.72240 0.60.725750.729070.732370.735650.738910.742150.745370.748570.751750.75490 0.70.758040.761150.764240.767300.770350.773370.776370.779350.782300.78524 0.80.788140.791030.793890.796730.799550.802340.805110.807850.810570.81327 0.90.815940.818590.821210.823810.826390.828940.831470.833980.836460.83891 1.00.841340.843750.846140.848490.850830.853140.855430.857690.859930.86214 1.10.864330.866500.868640.870760.872860.874930.876980.879000.881000.88298 1.20.884930.886860.888770.890650.892510.894350.896170.897960.899730.90147 1.30.903200.904900.906580.908240.909880.911490.913080.914660.916210.91774 1.40.919240.920730.922200.923640.925070.926470.927850.929220.930560.93189 1.50.933190.934480.935740.936990.938220.939430.940620.941790.942950.94408 1.60.945200.946300.947380.948450.949500.950530.951540.952540.953520.95449 1.70.955430.956370.957280.958180.959070.959940.960800.961640.962460.96327 1.80.964070.964850.965620.966380.967120.967840.968560.969260.969950.97062 1.90.971280.971930.972570.973200.973810.974410.975000.975580.976150.97670 2.00.977250.977780.978310.978820.979320.979820.980300.980770.981240.98169 2.10.982140.982570.983000.983410.983820.984220.984610.985000.985370.98574 2.20.986100.986450.986790.987130.987450.987780.988090.988400.988700.98899 2.30.989280.989560.989830.990100.990360.990610.990860.991110.991340.99158 2.40.991800.992020.992240.992450.992660.992860.993050.993240.993430.99361 2.50.993790.993960.994130.994300.994460.994610.994770.994920.995060.99520 2.60.995340.995470.995600.995730.995850.995980.996090.996210.996320.99643 2.70.996530.996640.996740.996830.996930.997020.997110.997200.997280.99736 2.80.997440.997520.997600.997670.997740.997810.997880.997950.998010.99807 2.90.998130.998190.998250.998310.998360.998410.998460.998510.998560.99861 3.00.998650.998690.998740.998780.998820.998860.998890.998930.998960.99900 3.10.999030.999060.999100.999130.999160.999180.999210.999240.999260.99929 3.20.999310.999340.999360.999380.999400.999420.999440.999460.999480.99950 3.30.999520.999530.999550.999570.999580.999600.999610.999620.999640.99965 3.40.999660.999680.999690.999700.999710.999720.999730.999740.999750.99976 3.50.999770.999780.999780.999790.999800.999810.999810.999820.999830.99983 3.60.999840.999850.999850.999860.999860.999870.999870.999880.999880.99989 3.70.999890.999900.999900.999900.999910.999910.999920.999920.999920.99992 3.80.999930.999930.999930.999940.999940.999940.999940.999950.999950.99995 3.90.999950.999950.999960.999960.999960.999960.999960.999960.999970.99997 4.00.999970.999970.999970.999970.999970.999970.999980.999980.999980.99998 z +0.00 +0.01 +0.02 +0.03 +0.04 +0.05 +0.06 +0.07 +0.08 +0.09
Қосымша жиынтық
Бұл кестеде статистикалық шаманың Z-ден үлкен болу ықтималдығы көрсетілген.
z +0.00+0.01+0.02+0.03+0.04 +0.05+0.06+0.07+0.08+0.09 0.00.500000.496010.492020.488030.484050.480060.476080.472100.468120.46414 0.10.460170.456200.452240.448280.444330.440380.436400.432510.428580.42465 0.20.420740.416830.412940.409050.405170.401290.397430.393580.389740.38591 0.30.382090.378280.374480.370700.366930.363170.359420.355690.351970.34827 0.40.344580.340900.337240.333600.329970.326360.322760.319180.315610.31207 0.50.308540.305030.301530.298060.294600.291160.287740.284340.280960.27760 0.60.274250.270930.267630.264350.261090.257850.254630.251430.248250.24510 0.70.241960.238850.235760.232700.229650.226630.223630.220650.217700.21476 0.80.211860.208970.206110.203270.200450.197660.194890.192150.189430.18673 0.90.184060.181410.178790.176190.173610.171060.168530.166020.163540.16109 1.00.158660.156250.153860.151510.149170.146860.144570.142310.140070.13786 1.10.135670.133500.131360.129240.127140.125070.123020.121000.119000.11702 1.20.115070.113140.111230.109350.107490.105650.103830.102040.100270.09853 1.30.096800.095100.093420.091760.090120.088510.086920.085340.083790.08226 1.40.080760.079270.077800.076360.074930.073530.072150.070780.069440.06811 1.50.066810.065520.064260.063010.061780.060570.059380.058210.057050.05592 1.60.054800.053700.052620.051550.050500.049470.048460.047460.046480.04551 1.70.044570.043630.042720.041820.040930.040060.039200.038360.037540.03673 1.80.035930.035150.034380.033620.032880.032160.031440.030740.030050.02938 1.90.028720.028070.027430.026800.026190.025590.025000.024420.023850.02330 2.00.022750.022220.021690.021180.020680.020180.019700.019230.018760.01831 2.10.017860.017430.017000.016590.016180.015780.015390.015000.014630.01426 2.20.013900.013550.013210.012870.012550.012220.011910.011600.011300.01101 2.30.010720.010440.010170.009900.009640.009390.009140.008890.008660.00842 2.40.008200.007980.007760.007550.007340.007140.006950.006760.006570.00639 2.50.006210.006040.005870.005700.005540.005390.005230.005080.004940.00480 2.60.004660.004530.004400.004270.004150.004020.003910.003790.003680.00357 2.70.003470.003360.003260.003170.003070.002980.002890.002800.002720.00264 2.80.002560.002480.002400.002330.002260.002190.002120.002050.001990.00193 2.90.001870.001810.001750.001690.001640.001590.001540.001490.001440.00139 3.00.001350.001310.001260.001220.001180.001140.001110.001070.001040.001003.10.000970.000940.000900.000870.000840.000820.000790.000760.000740.000713.20.000690.000660.000640.000620.000600.000580.000560.000540.000520.000503.30.000480.000470.000450.000430.000420.000400.000390.000380.000360.000353.40.000340.000320.000310.000300.000290.000280.000270.000260.000250.000243.50.000230.000220.000220.000210.000200.000190.000190.000180.000170.000173.60.000160.000150.000150.000140.000140.000130.000130.000120.000120.000113.70.000110.000100.000100.000100.000090.000090.000080.000080.000080.000083.80.000070.000070.000070.000060.000060.000060.000060.000050.000050.000053.90.000050.000050.000040.000040.000040.000040.000040.000040.000030.000034.00.000030.000030.000030.000030.000030.000030.000020.000020.000020.00002
This table gives a probability that a statistic is greater than Z, for large integer Z values. z +0+1+2+3+4 +5+6+7+8+9 05.00000 E −11.58655 E −12.27501 E −21.34990 E −33.16712 E −52.86652 E −79.86588 E −101.27981 E −126.22096 E −161.12859 E −19 107.61985 E −241.91066 E −281.77648 E −336.11716 E −397.79354 E −453.67097 E −516.38875 E −584.10600 E −659.74095 E −738.52722 E −81 202.75362 E 893.27928 E 981.43989 E 1072.33064 E 1171.39039 E 1273.05670 E 1382.47606 E 1497.38948 E 1618.12387 E 1733.28979 E 185 304.90671 E 1982.69525 E 2115.45208 E 2254.06119 E 2391.11390 E 2531.12491 E 2684.18262 E 2845.72557 E 3002.88543 E 3165.35312 E 333 403.65589 E 3509.19086 E 3688.50515 E 3862.89707 E 4043.63224 E 4231.67618 E 4422.84699 E 4621.77976 E 4824.09484 E 5033.46743 E 524 501.08060 E 5451.23937 E 5675.23127 E 5908.12606 E 6134.64529 E 6369.77237 E 6607.56547 E 6842.15534 E 7082.25962 E 7338.71741 E 759 601.23757 E 7846.46517 E 8111.24283 E 8378.79146 E 8652.28836 E 8922.19180 E 9207.72476 E 9491.00178 E 9774.78041 E 10078.39374 E 1037705.42304 E 10671.28921 E 10971.12771 E 11283.62960 E 11604.29841 E 11921.87302 E 12243.00302 E 12571.77155 E 12903.84530 E 13243.07102 E 1358
z +0.00+0.01+0.02+0.03+0.04 +0.05+0.06+0.07+0.08+0.09 0.000.500000.496010.492020.488030.484050.480060.476080.472100.468120.46414 0.100.460170.456200.452240.448280.444330.440380.436400.432510.428580.42465 0.200.420740.416830.412940.409050.405170.401290.397430.393580.389740.38591 0.300.382090.378280.374480.370700.366930.363170.359420.355690.351970.34827 0.400.344580.340900.337240.333600.329970.326360.322760.319180.315610.31207 0.500.308540.305030.301530.298060.294600.291160.287740.284340.280960.27760 0.600.274250.270930.267630.264350.261090.257850.254630.251430.248250.24510 0.700.241960.238850.235760.232700.229650.226630.223630.220650.217700.21476 0.800.211860.208970.206110.203270.200450.197660.194890.192150.189430.18673 0.900.184060.181410.178790.176190.173610.171060.168530.166020.163540.16109 1.000.158660.156250.153860.151510.149170.146860.144570.142310.140070.13786 1.100.135670.133500.131360.129240.127140.125070.123020.121000.119000.11702 1.200.115070.113140.111230.109350.107490.105650.103830.102040.100270.09853 1.300.096800.095100.093420.091760.090120.088510.086920.085340.083790.08226 1.400.080760.079270.077800.076360.074930.073530.072150.070780.069440.06811 1.500.066810.065520.064260.063010.061780.060570.059380.058210.057050.05592 1.600.054800.053700.052620.051550.050500.049470.048460.047460.046480.04551 1.700.044570.043630.042720.041820.040930.040060.039200.038360.037540.03673 1.800.035930.035150.034380.033620.032880.032160.031440.030740.030050.02938 1.900.028720.028070.027430.026800.026190.025590.025000.024420.023850.02330 2.000.022750.022220.021690.021180.020680.020180.019700.019230.018760.01831 2.100.017860.017430.017000.016590.016180.015780.015390.015000.014630.01426 2.200.013900.013550.013210.012870.012550.012220.011910.011600.011300.01101 2.300.010720.010440.010170.009900.009640.009390.009140.008890.008660.00842 2.400.008200.007980.007760.007550.007340.007140.006950.006760.006570.00639 2.500.006210.006040.005870.005700.005540.005390.005230.005080.004940.00480 2.600.004660.004530.004400.004270.004150.004020.003910.003790.003680.00357 2.700.003470.003360.003260.003170.003070.002980.002890.002800.002720.00264 2.800.002560.002480.002400.002330.002260.002190.002120.002050.001990.00193 2.900.001870.001810.001750.001690.001640.001590.001540.001490.001440.00139 3.000.001350.001310.001260.001220.001180.001140.001110.001070.001040.001003.100.000970.000940.000900.000870.000840.000820.000790.000760.000740.000713.200.000690.000660.000640.000620.000600.000580.000560.000540.000520.000503.300.000480.000470.000450.000430.000420.000400.000390.000380.000360.000353.400.000340.000320.000310.000300.000290.000280.000270.000260.000250.000243.500.000230.000220.000220.000210.000200.000190.000190.000180.000170.000173.600.000160.000150.000150.000140.000140.000130.000130.000120.000120.000113.700.000110.000100.000100.000100.000090.000090.000080.000080.000080.000083.800.000070.000070.000070.000060.000060.000060.000060.000050.000050.000053.900.000050.000050.000040.000040.000040.000040.000040.000040.000030.000034.000.000030.000030.000030.000030.000030.000030.000020.000020.000020.00002
z +0.00+0.01+0.02+0.03+0.04 +0.05+0.06+0.07+0.08+0.09 0.00.500000.496010.492020.488030.484050.480060.476080.472100.468120.46414 0.10.460170.456200.452240.448280.444330.440380.436400.432510.428580.42465 0.20.420740.416830.412940.409050.405170.401290.397430.393580.389740.38591 0.30.382090.378280.374480.370700.366930.363170.359420.355690.351970.34827 0.40.344580.340900.337240.333600.329970.326360.322760.319180.315610.31207 0.50.308540.305030.301530.298060.294600.291160.287740.284340.280960.27760 0.60.274250.270930.267630.264350.261090.257850.254630.251430.248250.24510 0.70.241960.238850.235760.232700.229650.226630.223630.220650.217700.21476 0.80.211860.208970.206110.203270.200450.197660.194890.192150.189430.18673 0.90.184060.181410.178790.176190.173610.171060.168530.166020.163540.16109 1.00.158660.156250.153860.151510.149170.146860.144570.142310.140070.13786 1.10.135670.133500.131360.129240.127140.125070.123020.121000.119000.11702 1.20.115070.113140.111230.109350.107490.105650.103830.102040.100270.09853 1.30.096800.095100.093420.091760.090120.088510.086920.085340.083790.08226 1.40.080760.079270.077800.076360.074930.073530.072150.070780.069440.06811 1.50.066810.065520.064260.063010.061780.060570.059380.058210.057050.05592 1.60.054800.053700.052620.051550.050500.049470.048460.047460.046480.04551 1.70.044570.043630.042720.041820.040930.040060.039200.038360.037540.03673 1.80.035930.035150.034380.033620.032880.032160.031440.030740.030050.02938 1.90.028720.028070.027430.026800.026190.025590.025000.024420.023850.02330 2.00.022750.022220.021690.021180.020680.020180.019700.019230.018760.01831 2.10.017860.017430.017000.016590.016180.015780.015390.015000.014630.01426 2.20.013900.013550.013210.012870.012550.012220.011910.011600.011300.01101 2.30.010720.010440.010170.009900.009640.009390.009140.008890.008660.00842 2.40.008200.007980.007760.007550.007340.007140.006950.006760.006570.00639 2.50.006210.006040.005870.005700.005540.005390.005230.005080.004940.00480 2.60.004660.004530.004400.004270.004150.004020.003910.003790.003680.00357 2.70.003470.003360.003260.003170.003070.002980.002890.002800.002720.00264 2.80.002560.002480.002400.002330.002260.002190.002120.002050.001990.00193 2.90.001870.001810.001750.001690.001640.001590.001540.001490.001440.00139 3.00.001350.001310.001260.001220.001180.001140.001110.001070.001040.001003.10.000970.000940.000900.000870.000840.000820.000790.000760.000740.000713.20.000690.000660.000640.000620.000600.000580.000560.000540.000520.000503.30.000480.000470.000450.000430.000420.000400.000390.000380.000360.000353.40.000340.000320.000310.000300.000290.000280.000270.000260.000250.000243.50.000230.000220.000220.000210.000200.000190.000190.000180.000170.000173.60.000160.000150.000150.000140.000140.000130.000130.000120.000120.000113.70.000110.000100.000100.000100.000090.000090.000080.000080.000080.000083.80.000070.000070.000070.000060.000060.000060.000060.000050.000050.000053.90.000050.000050.000040.000040.000040.000040.000040.000040.000030.000034.00.000030.000030.000030.000030.000030.000030.000020.000020.000020.00002
This table gives a probability that a statistic is greater than Z, for large integer Z values. z +0+1+2+3+4 +5+6+7+8+9 05.00000 E −11.58655 E −12.27501 E −21.34990 E −33.16712 E −52.86652 E −79.86588 E −101.27981 E −126.22096 E −161.12859 E −19 107.61985 E −241.91066 E −281.77648 E −336.11716 E −397.79354 E −453.67097 E −516.38875 E −584.10600 E −659.74095 E −738.52722 E −81 202.75362 E 893.27928 E 981.43989 E 1072.33064 E 1171.39039 E 1273.05670 E 1382.47606 E 1497.38948 E 1618.12387 E 1733.28979 E 185 304.90671 E 1982.69525 E 2115.45208 E 2254.06119 E 2391.11390 E 2531.12491 E 2684.18262 E 2845.72557 E 3002.88543 E 3165.35312 E 333 403.65589 E 3509.19086 E 3688.50515 E 3862.89707 E 4043.63224 E 4231.67618 E 4422.84699 E 4621.77976 E 4824.09484 E 5033.46743 E 524 501.08060 E 5451.23937 E 5675.23127 E 5908.12606 E 6134.64529 E 6369.77237 E 6607.56547 E 6842.15534 E 7082.25962 E 7338.71741 E 759 601.23757 E 7846.46517 E 8111.24283 E 8378.79146 E 8652.28836 E 8922.19180 E 9207.72476 E 9491.00178 E 9774.78041 E 10078.39374 E 1037705.42304 E 10671.28921 E 10971.12771 E 11283.62960 E 11604.29841 E 11921.87302 E 12243.00302 E 12571.77155 E 12903.84530 E 13243.07102 E 1358